From Pendulums to p-Adic Numbers - A Philosophy of Mathematics

For quite some time I have been working on an essay on the philosophy of mathematics. It's gone through many changes since I first started jotting notes on the commute and in various cafes around London, and bears no resemblance to anything I thought I would write when I started. It isn't complete and probably never will be, since there are always more insights and examples to add. It does have most of the philosophical points I want to make, and talks about most of the maths I feel even half-way competent to discuss. So I'm going to make it available for whoever needs some light entertainment. It will get updated from time to time.

The link is here (link)

It's an attempt to answer these questions:

How is it mathematical techniques and tools are so suited to describe physical processes? 

How do mathematical concepts work?


What kinds of knowledge does mathematics provide?


How do we know that a theory does not harbour fatal inconsistencies? 

How do mathematicians get and develop their ideas?


How do we judge the value of a technique, theorem or subject? 

What constitutes progress in maths?

It proceeds through discussions of these issues in the context of differential equations, functional analysis, infinity, functions, numerical analysis and recursive functions, and the various types of numbers, from the counting numbers to the p-adics. There's a discussion of axiomatics and model theory and a brief look at category theory; the way mathematical ideas are structured and what mathematical knowledge is (epistemology); how we might appraise different mathematical theories (methodology); and what constitutes progress and then a discussion of how to get ideas and solve problems (heuristics).

What there isn't is detailed presentations and rebuttals of existing philosophies of mathematics, what I've called the “where Smith mistakes Jones’ summary of Brown’s critique of Frege’s

misunderstanding of Kant” school of scholarly discussion. 

Rigging

 

You know which boat this is, and where it's located. Worth clicking through to get a better view of all those cables and ropes, none of which are called "cables" or "ropes" by Real Sailors, but then. I'm not a Real Sailor.

Charlton House

 Most of it is open to the public, but sadly there's no historic furniture, art or decoration there. It's a ten-minute walk up the hill from Charlton station, and worth an amble around the park, a cup of coffee and slice of Victoria cake in the cafe. 

Negative Space, London Bridge Station

All the Kool Photographers talk about using "negative space", but I always thought it meant they exposed those part of the picture incorrectly. But this one seems to work.

 

Bleak Mid-Winter Suburbia

It's not enough to get out for a daily walk. The walk needs to be pleasant, or at least neutral, to look at. Hedges on country lanes, with an occasional glimpse across a valley, or perhaps a path across a flat moor, or maybe even along a canal. Not round the outside of an industrial estate. But we make do and carry on.

 

One Wall of the Walled Garden, Golders Hill Park

 

Golders Hill Park is a couple of stops up the hill from the station. It's well worth the visit.

Room Resonances

Room resonances are a real thing, but... a) the wavelength has to match the room dimension almost exactly.

While it looks as though there are "as many notes as we want", in Western music there are only 88 notes. But not really. There are actually 12 fundamental notes - starting with A₀ at 27.5 Hz and ending at A♭₁ at 51.91 Hz. Double those frequencies to get the next octave; double again to get the next; and so on until reaching C₈ at 4186 Hz.

So a room that supports a standing wave (resonance) at, say, C₂ 65.41 Hz, will support standing waves at all the other C's as well. The sound will be quieter with each jump up or down of an octave. However, people only worry about bass resonances. That's because notes below a limit that varies with the room, are non-directional, appearing, as it were, at once everywhere in the room. (Above that limit, the notes become directional, which is how you ears tell you that the drums are right in the middle of your speakers.) Think of the bass notes as being produced in the middle of the room and going in all directions. If one of the dimensions of the room fits the note, and if there isn't soft furniture in the way, up pops a resonance.

If you're really unlucky you might get three different resonances: floor-to-ceiling, side-to-side, front-to-back. Highly unlikely, but possible. Chances are you will get one. There won't be others, unless your room changes dimension somewhere (sloping walls or ceiling?). Most people will get one. And that's it.

My listening room is 2.5m high, so a slightly out of tune C♯₃ / D♭₃ of 138.6 Hz will cause a stomach-churning resonance. Here's the thing: the 3-octave is used for effect, not for carrying the tune. That's usually done an octave higher where resonances don't happen. Bass players famously "play the root note" (unless they are Jaco Pastorius or Jack Bruce), and C♯₃ / D♭₃ (or C♯₂ / D♭₂) are not the most frequent root notes. Also, the instrument would need to be slightly out of tune to make my room react. That's why it happens so infrequently.

That doesn't mean I don't get quieter and louder patches if I move the subwoofer around. Very much so: interference isn't resonance. Its current position was chosen because it produced the most uniform level throughout the room. It's very un-nerving moving from one chair to another and suddenly hearing more bass.

Anyway, here's a list of the notes most likely to cause resonances, along with the wavelength. Measure the room (wall-to-wall, ceiling to floor. You can ignore diagonals because corners create bass boost, but do not create standing waves) and if any of those three numbers are within 0.02m (20mm) or so (depends on how reflective the material is), you will likely get resonances

D♭₃      2.47m 

C₃          2.62m / 130Hz 

B₂          2.78m 

B♭₂      2.94m 

A₂          3.12m / 110Hz 

A♭₂      3.30m 

G₂          3.5m 

F♯₂       3.71m 

F₂           3.93m 

E₂           4.16m 

E♭₂       4.41m 

D₂          4.67m 

D♭₂      4.94m 

C₂          5.24m 

B₁          5.56m 

B♭₁      5.88m 

A₁           6.24m 55Hz 

A♭₁       6.6m 

G₁          7.0m 

F♯₁        7.42m 

F₁           7.86m 

E₁           8.32m 

E♭₁       8.82m 

D₁          9.34 

D♭₁      9.9m 

C₁         10.48m 

B₀         11.12m 

B♭₀     11.76m 

A₀         12.48m 27.5Hz

How do you stop a resonance? Only big, obtrusive, and expensive bass traps made of materials sourced in an Ardennes forest and hand-assembled by elves in a workshop outside Dusseldorf will do the trick... it says here on the PR handout.

Resonances result from room dimensions. So change the dimensions of the room. No builders needed. Nice full shelves full of absorbent things: paperbacks are always good, just don't line them up precisely. LP's or big art hardbacks may not be a good idea if the resonances are at higher frequencies. This will work for side-to-side or back-to-front resonances, but floor-to-ceiling you are pretty much stuck with. Unless you put nice thick carpet in everywhere, which will damp it a little.